Limits, Continuity & Differentiability
Limit at infinity involving 6th power expressions
nta_pyq_2023_jan
Grade 12
Question:
\lim_{x \to \infty} \frac{\left(\sqrt{3x+1}+\sqrt{3x-1}\right)^6 + \left(\sqrt{3x+1}-\sqrt{3x-1}\right)^6}{\left(x+\sqrt{x^2-1}\right)^6 + \left(x-\sqrt{x^2-1}\right)^6} \cdot x^3
is equal to 9
is equal to 27
does not exist
is equal to \frac{27}{2}
Step-by-Step Solution
Key Concept: Divide numerator and denominator by x^3 and x^6 respectively and take leading terms as x \to \infty.
As x \to \infty, numerator \sim (2\sqrt{3x})^6 = (2\sqrt{3})^6 x^3 and denominator \sim (2x)^6 / x^3... = (2\sqrt{3})^6/(2^6) = 27.
Correct Answer: 2